The first term in (REF ) is deterministic and can be approximated by
\( Q_{1}(M):=\mu _{M^{*},N^{*},L^{*}} - \mu _{M,N^{*},L^{*}}\)  with large enough \( L^{*} \) , \( M^{*} \)  and \( N^{*}. \)  The variability of the second, zero mean, stochastic term can be measured by \(\sqrt{\operatorname{Var}_{\operatorname{P}^{\otimes M}} \left[V_{M} \right]} \)  which in turn can be estimated by \( Q_{2}(M):=\sqrt{\vartheta _{M,N^{*},L^{*}}} \) , due to (). The standard deviation of \(V_{M}-V_{M,N}\)  for any \( M \)  can be approximated by \( Q_{3}(N)=\sigma _{M^{*},N} /\sqrt{N} \) .
In our simulation study we take \( N^{*}=1000000, \, L^{*}=500,\, M^{*}=10000 \)  and obtain \( \bar{V}\approx \mu _{M^{*},N^{*},L^{*}}=7.96 \)  (note that \( V^{*}=8.07 \)  according to
[1]}). In the left-hand side of Figure REF  we plot both quantities \( Q_{1}(M) \)  and
\( Q_{2}(M) \)  as functions of \( M. \)  Note that \( Q_{2}(M) \)  dominates \( Q_{1}(M) \) , especially for large \( M. \) 
Hence, by comparing \( Q_{2}(M) \)  with \( Q_{3}(N) \)  and approximately solving the equation \( Q_{2}(M)=Q_{3}(N) \)  in \( N \) , one can infer on the optimal relation between \( M \)  and \( N \) .
In Figure REF  (on the right-hand side) the resulting empirical relation is depicted by crosses. Additionally, we plotted two benchmark curves \( N=M^{4/3} \)  and \( N=M^{4.5/3} \) .
As one can see the choice \( M=N^{3/4} \)  is likely to be sufficient in this situation since it always leads to the inequality \( Q_{1}(M)+\sigma Q_{2}(M)\le \sigma Q_{3}(N) \)  for any \( \sigma >1. \)  As a consequence, for \( M=N^{3/4} \)  and any \( N \) , \( \bar{V} \) 
lies with high probability in the interval \( [\mu _{M,N,L^{*}}-\sigma Q_{3}(N),\mu _{M,N,L^{*}}+\sigma Q_{3}(N)],\)  provided that \( \sigma \)  is large enough.
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